Smooth, invariant orthonormal basis for singular potential Schroedinger operators

• URL: http://arxiv.org/abs/2308.07059v1
• Date: Mon, 14 Aug 2023 10:38:26 GMT
• Title: Smooth, invariant orthonormal basis for singular potential Schroedinger operators
• Authors: J. Neuser, T. Thiemann
• Abstract summary: In a recent contribution we showed that there exists a smooth, dense domain for singular potential Schr"odinger operators on the real line. inner products between basis elements of that domain were shown to be easily computable analytically. A task left open was to construct an orthonormal basis from elements of that domain by using Gram-Schmidt orthonormalisation.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In a recent contribution we showed that there exists a smooth, dense domain for singular potential Schr\"odinger operators on the real line which is invariant under taking derivatives of arbitrary order and under multiplication by positive and negative integer powers of the coordinate. Moreover, inner products between basis elements of that domain were shown to be easily computable analytically. A task left open was to construct an orthonormal basis from elements of that domain by using Gram-Schmidt orthonormalisation. We perform that step in the present manuscript. We also consider the application of these methods to the positive real line for which one can no longer perform the integrals analytically but for which one can give tight analytical estimates.

Related papers

• Entrywise error bounds for low-rank approximations of kernel matrices [55.524284152242096]
  We derive entrywise error bounds for low-rank approximations of kernel matrices obtained using the truncated eigen-decomposition. A key technical innovation is a delocalisation result for the eigenvectors of the kernel matrix corresponding to small eigenvalues. We validate our theory with an empirical study of a collection of synthetic and real-world datasets.
  arXiv  Detail & Related papers  (2024-05-23T12:26:25Z)
• A metaplectic perspective of uncertainty principles in the Linear Canonical Transform domain [0.0]
  We derive Heisenberg uncertainty principles for pairs of Linear Canonical Transforms of a given function. We also propose a new quadratic phase-space distribution, which represents a signal along two intermediate directions in the time-frequency plane.
  arXiv  Detail & Related papers  (2024-05-17T09:26:48Z)
• Curvature-Independent Last-Iterate Convergence for Games on Riemannian Manifolds [77.4346324549323]
  We show that a step size agnostic to the curvature of the manifold achieves a curvature-independent and linear last-iterate convergence rate. To the best of our knowledge, the possibility of curvature-independent rates and/or last-iterate convergence has not been considered before.
  arXiv  Detail & Related papers  (2023-06-29T01:20:44Z)
• Remarks on the quasi-position representation in models of generalized uncertainty principle [0.0]
  This note aims to elucidate certain aspects of the quasi-position representation frequently used in the investigation of one-dimensional models. We focus on two key points: (i) Contrary to recent claims, the quasi-position operator can possess physical significance even though it is non-Hermitian, and (ii) in the quasi-position representation, operators associated with the position behave as a derivative operator on the quasi-position coordinate.
  arXiv  Detail & Related papers  (2023-06-20T11:46:56Z)
• Properties of a smooth, dense, invariant domain for singular potential Schroedinger operators [0.0]
  We show that relevant matrix elements and inner products can be computed analytically in closed form. This provides the required data for an analytical Gram-Schmid orthonormalisation.
  arXiv  Detail & Related papers  (2023-05-11T10:54:14Z)
• The eigenvalues and eigenfunctions of the toroidal dipole operator in a mesoscopic system [0.0]
  We find the quantization rules for the eigenvalues, which are essential for describing measurements of $hatT_3$. While these kernels appear to be problematic at first glance due to singularities, they can actually be used in practical computations.
  arXiv  Detail & Related papers  (2022-03-19T17:49:20Z)
• On computing bound states of the Dirac and Schr\"odinger Equations [0.0]
  We show that by changing the parameter, we can always find the bound states that satisfy the original equations and are normalizable. While for the non-relativistic equations these properties may not be surprising, it is remarkable that the same holds for the relativistic equations.
  arXiv  Detail & Related papers  (2021-07-05T20:00:20Z)
• Minimax Estimation of Linear Functions of Eigenvectors in the Face of Small Eigen-Gaps [95.62172085878132]
  Eigenvector perturbation analysis plays a vital role in various statistical data science applications. We develop a suite of statistical theory that characterizes the perturbation of arbitrary linear functions of an unknown eigenvector. In order to mitigate a non-negligible bias issue inherent to the natural "plug-in" estimator, we develop de-biased estimators.
  arXiv  Detail & Related papers  (2021-04-07T17:55:10Z)
• Models of zero-range interaction for the bosonic trimer at unitarity [91.3755431537592]
  We present the construction of quantum Hamiltonians for a three-body system consisting of identical bosons mutually coupled by a two-body interaction of zero range. For a large part of the presentation, infinite scattering length will be considered.
  arXiv  Detail & Related papers  (2020-06-03T17:54:43Z)
• Quantum Geometric Confinement and Dynamical Transmission in Grushin Cylinder [68.8204255655161]
  We classify the self-adjoint realisations of the Laplace-Beltrami operator minimally defined on an infinite cylinder. We retrieve those distinguished extensions previously identified in the recent literature, namely the most confining and the most transmitting.
  arXiv  Detail & Related papers  (2020-03-16T11:37:23Z)
• SLEIPNIR: Deterministic and Provably Accurate Feature Expansion for Gaussian Process Regression with Derivatives [86.01677297601624]
  We propose a novel approach for scaling GP regression with derivatives based on quadrature Fourier features. We prove deterministic, non-asymptotic and exponentially fast decaying error bounds which apply for both the approximated kernel as well as the approximated posterior.
  arXiv  Detail & Related papers  (2020-03-05T14:33:20Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.